Subduction dynamics as revealed by trench migration

Transcription

1 TECTONICS, VOL. 27,, doi: /2007tc002212, 2008 Subduction dynamics as revealed by trench migration Serge Lallemand, 1 Arnauld Heuret, 1 Claudio Faccenna, 2 and Francesca Funiciello 2 Received 25 September 2007; revised 5 March 2008; accepted 28 April 2008; published 28 June [1] New estimates of trench migration rates allow us to address the dynamics of trench migration and backarc strain. We show that trench migration is primarily controlled by the subducting plate velocity V sub, which largely depends on its age at the trench. Using the hot and weak arc to back-arc region as a strain sensor, we define neutral arcs characterized by the absence of significant strain, meaning places where the forces (slab pull, bending, and anchoring) almost balance along the interface between the plates. We show that neutral subduction zones satisfy the kinematic relation between trench and subducting plate absolute motions: V t =0.5V sub 2.3 (in cm a 1 ) in the HS3 reference frame. Deformation occurs when the velocity combination deviates from kinematic equilibrium. Balancing the torque components of the forces acting at the trench indicates that stiff (old) subducting plates facilitate trench advance by resisting bending. Citation: Lallemand, S., A. Heuret, C. Faccenna, and F. Funiciello (2008), Subduction dynamics as revealed by trench migration, Tectonics, 27,, doi: / 2007TC Introduction [2] A trench is the geographic location of a subduction plate boundary at a given time. This location may change over time as a result of global plate tectonics. Carlson and Melia [1984] stated that back-arc tectonics is controlled principally by the difference in absolute motion between the overriding plate and the migration of the hinge of the downgoing plate. In this study, we test the hypothesis that trench migration is dependent on lower plate parameters in light of updated kinematic data. Since volcanic arcs or back arcs are generally hot and weak [e.g., Currie and Hyndman, 2006], we use this area as a sensor, represented by a spring in Figure 1, to determine the stress transmitted from one plate to the other. The lack of strain is supposed to indicate limited deviatoric stress, i.e., low stress transmission from one plate to the other through the interface between them. If no significant deformation occurs in the arc-back-arc region (normal component of deformation rate v d 0, neutral regime), then the subduction rate (v s ) equals the 1 UMR 5573, CNRS, Laboratoire Géosciences Montpellier, Montpellier II Université, Montpellier, France. 2 Dipartimento di Scienze Geologiche, Università degli Studi Roma TRE, Roma, Italy. Copyright 2008 by the American Geophysical Union /08/2007TC convergence rate between the major plates (v c ), and the velocity of the trench (V t ) equals the velocity of the upper plate (V up ). We use the respective plate motions to explore the kinematic conditions required for the absence of upper plate strain. Trench migration in such neutral subduction zones provides useful information about the balance of forces between the subducting plate and the mantle without significant interaction with the overriding plate. In contrast, if deformation occurs (v d 6¼ 0), the subduction rate will depend on both the convergence velocity between the major plates (v c ) and the deformation velocity (v d ). We can therefore explore the kinematic criteria that characterize subduction zones when the trench moves spontaneously, i.e., neutral subduction zones, and those experiencing active shortening or spreading and ultimately infer the dynamic equilibrium of forces along the interface between the plates. 2. Data [3] We use the global database already published in work by Heuret and Lallemand [2005] and Lallemand et al. [2005], which consists of transects every 2 of latitude/ longitude (200 km) along subduction zones. From this database, we have excluded the transects where continental crust, ridges, or plateaus subduct and those with slabs narrower than 500 km. This exclusion focused our study on 166 transects (see Table 1). We use the normal component of velocity. Upper plate strain is determined from focal mechanisms of earthquakes occurring at a depth of less than 40 km from the surface of the upper plate, far from the subduction interface. We have simplified the seven strain classes described by Heuret and Lallemand [2005] into three: E3 and E2 become extensional (rifting or spreading); E1, 0, and C1 become neutral (no significant deformation or strike slip); and C2 and C3 become compressional (shortening). [4] We distinguish between v s, V up, V t, and V sub to describe the velocities of subduction, of the upper plate, the trench, and the subducting plate, respectively. We estimate V t by subtracting the deformation velocity v d as estimated by geodetic measurements from V up, neglecting erosion and accretion at the toe of the margin s wedge. Velocity is estimated using three different reference frames: NNR [DeMets et al., 1994; Gripp and Gordon, 2002], SB04 [Steinberger et al., 2004; Becker, 2006], and HS3 [Gripp and Gordon, 2002]. These three models span the whole range of net rotation of the lithosphere with respect to the mantle important to the kinematic relationships [e.g., Doglioni et al., 2007]. Finally, because our study is mainly focused on Pacific subduction zones, we prefer a reference frame that includes Pacific hot spot analysis, such as the HS3. 1of15

2 Figure 1. Definition of velocities in a schematic subduction zone. A weak arc is compared with a spring that deforms at a rate v d (positive for spreading). And v c is the relative plate convergence rate. Subduction rate v s is the sum of convergence rate v c and deformation rate v d. V up, V t, and V sub are the absolute upper plate, trench, and subducting plate motions counted positive landward. Upper case velocities are absolute, and lower case velocities are relative Trench Migration [5] Quantitative estimates of trench motion are done in a given reference frame. For a long time, scientists considered that trenches should migrate spontaneously seaward with respect to a passive underlying mantle, so-called trench rollback, as a consequence of the downward pull exerted by the excess mass of slabs in the mantle [Molnar and Atwater, 1978; Dewey, 1980; Garfunkel et al., 1986]. Back arc spreading was thus associated with trench rollback [e.g., Molnar and Atwater, 1978]. However, some authors noticed early on that trenches might advance toward the arc [e.g., Carlson and Melia, 1984; Jarrard, 1986; Otsuki, 1989]. Recently, updated estimates of trenches motion have become available, based on a new global database of all oceanic subduction zones [Heuret and Lallemand, 2005]. Table 2 gives estimates for three different reference frames. Depending on the reference frame, the mean net rotation of the lithosphere with respect to the mantle varies from 0 (NNR) to 3.8 cm a 1 (HS3), resulting in a global tendency for slabs to retreat (rollback) at mean rates varying from 1.1 ± 3.0 (NNR) to 0.6 ± 3.0 cm a 1 (HS3). The maximum rate of trench rollback is in the northern Tonga region (12.5 (SB04) to 14.5 cm a 1 (HS3)). The main effect of net rotation is found in the ratio between advancing and retreating trenches, even if retreating rates are faster on average than advancing ones. Thirty percent of trenches in SB04 are prograding toward arcs. This number becomes 39% using NNR and 53% using HS3 reference frame. In HS3 reference frame, 50% of modern back-arc basins open above advancing trenches, and more generally, back-arc deformation scales with the absolute motion of the upper plate [Heuret and Lallemand, 2005]. In addition, trench (hinge) velocity scales with (subducting) plate velocity in a retreating mode for low plate velocities and an advancing mode for fast ones (Figure 2) [Heuret, 2005; Faccenna et al., 2007]. This correlation, although less pronounced, also holds for NNR and SB04 reference models [Funiciello et al., 2008]. [6] This global kinematic relationship has been poorly investigated, possibly because the dynamics of advancing trenches have never been accurately reproduced. Recently, Bellahsen et al. [2005], Schellart [2005], Funiciello et al. [2008], and Di Giuseppe et al. [2008] observed that both advancing and retreating subduction can be obtained in the laboratory, depending on physical parameters such as plate thickness, stiffness, width, velocity, mantle viscosity, and tank height. In addition, the experimental scaling relationships between plate and trench velocity have been used to predict the global trench and plate velocity using the radius of curvature and the age of the plate at the trench [Faccenna et al., 2007]. [7] To summarize, trench migrations occur in both directions (arcward or seaward) for modern subduction zones. In most cases, trench velocities correlate with subducting plate velocities: retreating and advancing modes are preferentially observed for slow and fast subducting plates, respectively (Figure 2) Relationship Between Upper and Subducting Plate Velocities and Slab Age at Trench [8] In this study, on the basis of modern subduction zone observations, we consider the subducting plate velocity results from a 3-D dynamic equilibrium between negative buoyancy of the slab and the resisting forces both in the lithosphere (bending and shear) and in the mantle [Forsyth and Uyeda, 1975]. We also consider that the trench migration V t mainly depends on V sub, as shown in Figure 2. Conversely, the main overriding plate velocity V up also results from a 3-D balance of forces, which can be far-field with respect to the subduction zone. For example, the northwestern motion of the Philippine Sea plate with respect to Eurasia is largely due to the pull exerted by the Ryukyu slab and is not influenced as much by its coupling with the Pacific plate at the Izu-Bonin-Mariana trench, by lateral density, or by topographic variations [Pacanovsky et al., 1999]. On a plot of the motion of the main overriding plate V up versus that of the subducting plate V sub for modern oceanic subduction zones in the HS3 reference frame (Figure 3), Heuret [2005] observed that these parameters are correlated. [9] First of all, 98% of normal components of convergence velocities v c between major plates are between 0 and 10 cm a 1. Second, all transects characterized by subducting oceanic plates older than 70 Ma are associated with retreating overriding plates. These two first-order observations led Heuret [2005] to propose the following empirical law: v c = V sub V up =5±5cma 1. This can be reached either by an old and fast lower plate subducting beneath a retreating overriding plate (e.g., Izu-Bonin-Mariana) or by young and slow plate subducting beneath an advancing upper plate (e.g., Andes, see insert of Figure 3). Changing the reference frame distorts the distribution of velocities slightly and thus the regression slope but does not alter the 2of15

3 Table 1. Data Set for 166 Transects Across Oceanic Subduction Zones a Reference Frame Transects HS3 SB04 NNR V subn V upn V tn V subn V upn V tn V subn V upn V tn Slab Age at Trench Upper Plate Strain Main Upper Plate ANDA Sunda ANDA Sunda ANDA Sunda ANDA Sunda ANDA Sunda ANDA Sunda SUM Sunda SUM Sunda SUM Sunda SUM Sunda SUM Sunda SUM Sunda JAVA Sunda JAVA Sunda JAVA Sunda JAVA Sunda JAVA Sunda JAVA Sunda JAVA Sunda LUZ Phil. Sea LUZ Phil. Sea LUZ Phil. Sea LUZ Phil. Sea BAT Phil. Sea PHIL Sunda PHIL Sunda PHIL Sunda PHIL Sunda PHIL Sunda PHIL Sunda RYUS Eurasia RYUN Eurasia RYUN Eurasia RYUN Eurasia RYUN Eurasia NAN Amur NAN Amur NAN Amur SMAR Phil. Sea SMAR Phil. Sea SMAR Phil. Sea SMAR Phil. Sea SMAR Phil. Sea NMAR Phil. Sea NMAR Phil. Sea NMAR Phil. Sea NMAR Phil. Sea IZU Phil. Sea IZU Phil. Sea IZU Phil. Sea IZU Phil. Sea JAP Amur JAP Amur JAP Amur JAP Amur SKOUR N-Amer. SKOUR N-Amer. SKOUR N-Amer. SKOUR N-Amer. SKOUR N-Amer. NKOUR N-Amer. NKOUR N-Amer. NKOUR N-Amer. KAM N-Amer. KAM N-Amer. W_ALE N-Amer. W_ALE N-Amer. 3of15

4 Table 1. (continued) Reference Frame Transects HS3 SB04 NNR V subn V upn V tn V subn V upn V tn V subn V upn V tn Slab Age at Trench Upper Plate Strain Main Upper Plate C_ALE N-Amer. C_ALE N-Amer. C_ALE N-Amer. C_ALE N-Amer. C_ALE N-Amer. C_ALE N-Amer. E_ALE N-Amer. E_ALE N-Amer. E_ALE N-Amer. E_ALE N-Amer. E_ALE N-Amer. W_ALA N-Amer. W_ALA N-Amer. W_ALA N-Amer. W_ALA N-Amer. W_ALA N-Amer. E_ALA N-Amer. E_ALA N-Amer. E_ALA N-Amer. E_ALA N-Amer. E_ALA N-Amer. CASC N-Amer. CASC N-Amer. CASC N-Amer. CASC N-Amer. CASC N-Amer. MEX N-Amer. MEX N-Amer. MEX N-Amer. MEX N-Amer. MEX N-Amer. MEX N-Amer. COST Carribean COST Carribean COST Carribean COST Carribean COST Carribean COL S-Amer. COL S-Amer. COL S-Amer. PER S-Amer. PER S-Amer. PER S-Amer. PER S-Amer. PER S-Amer. PER S-Amer. NCHI S-Amer. NCHI S-Amer. NCHI S-Amer. NCHI S-Amer. NCHI S-Amer. NCHI S-Amer. JUAN S-Amer. JUAN S-Amer. SCHI S-Amer. SCHI S-Amer. SCHI S-Amer. SCHI S-Amer. SCHI S-Amer. TRI S-Amer. TRI S-Amer. TRI S-Amer. TRI S-Amer. PAT S-Amer. PAT S-Amer. PAT S-Amer. BARB Carribean 4of15

5 Table 1. (continued) Reference Frame HS3 SB04 NNR Transects V subn V upn V tn V subn V upn V tn V subn V upn V tn Slab Age at Trench Upper Plate Strain Main Upper Plate BARB Carribean ANTI Carribean ANTI Carribean ANTI Carribean PORTO Carribean PORTO Carribean PORTO Carribean SAND Scotia SAND Scotia SAND Scotia SAND Scotia SAND Scotia SAND Scotia MAK Eurasia MAK Eurasia MAK Eurasia KER Australia KER Australia KER Australia KER Australia KER Australia TONG Australia TONG Australia TONG Australia TONG Australia SHEB Pacific SHEB Pacific NHEB Pacific NHEB Pacific BRET n-bism. BRET n-bism. BRET n-bism. a All velocities are expressed in mm a 1. V subn, V upn, and V tn (simply called V sub, V up, and V t in the text) are the normal components of the subducting plate, upper plate, and trench in the three reference frames NNR, SB04, and HS3. Age is given in Ma. Back-arc strain is 1 for extension, 0 for neutral, and 1 for compression as deduced from the study of focal mechanisms of earthquakes (see Heuret and Lallemand [2005] for further details). Phil. Sea, Philippine Sea; N-Amer., North America; S-Amer., South America; N-Bism., north Bismarck. empirical law regarding v c or the fact that old slabs are correlated with fast subducting plates Relationship Between the Combination of Upper and Subducting Plate Velocities and Upper Plate Strain [10] Given the conclusions that V up and V sub are not distributed randomly but are combined, such as V sub V up +5cma 1, we can now examine their combination with respect to upper plate strain: extensional, neutral, or compressional. We have plotted the results for the 166 transects in the HS3 reference frame (Figure 4a) but also in the two other reference frames discussed in section 2.1 (NNR, SB04) (Figures 4b and 4c) for comparison. [11] Let us first discuss the results in the HS3 reference frame. Obviously, the (V up and V sub ) combinations of velocities are not randomly distributed with respect to the associated upper plate strain. We distinguish three elongated domains, all three of which possess positive slopes (Figures 4a and 4b): [12] 1. The first is a domain of kinematic equilibrium where neither compressional nor extensional deformation is observed. The equation of the regression line for neutral transects (the neutral line) is V up ¼ 0:5V sub 2:3 or V sub 2V up ¼ 4:6 in cm a 1 : Along this line, the velocity of the trench (V t ) equals the velocity of the upper plate (V up ) because no deformation occurs: v d = 0. We can thus write V t ¼ 0:5V sub 2:3 only for those subduction zones with no upper plate deformation. [13] 2. The second is a domain where active shortening is observed in the overriding plate. Transects in this domain satisfy the relation V up <0.5V sub 2.3. Compression within the overriding plate occurs not only for trenchward moving upper plates, but also for retreating main overriding plate, like in NE Japan. 5of15

6 Table 2. Characteristics of Trench Motions in Three Reference Frames Reference Frame NNR-NUVEL1A [Gripp and Gordon, 2002] SB04 [Steinberger et al., 2004] HS3-NUVEL1A [Gripp and Gordon, 2002] Selected hot spots no hot spots Indo-Atlantic hot spots Pacific hot spots Age of hot spots - 80 Ma present 5.8 Ma present Mean/maximum net rotation 0 by definition 1.4/1.8 cm a 1 3.8/4.9 cm a 1 Mean trench motion 1.1 ± 3.0 cm a 1 (rollback) 0.9 ± 3.0 cm a 1 (rollback) 0.6 ± 4.0 cm a 1 (rollback) Maximum rate of trench rollback 12.7 cm a cm a cm a 1 Maximum rate of trench advance 5.1 cm a cm a cm a 1 Rollback/advance transects ratio [14] 3. The third is a domain where active extension is observed in the overriding plate. Transects in this domain satisfy the relation V up >0.5V sub 2.3. We observe a single exception, i.e., the four transects across the New Hebrides trench, discussed in section 3. [15] Figure 5 illustrates three specific combinations of velocities along the neutral line where v d = 0, and thus V t = V up. We first consider a fixed subducting plate: the upper plate needs to advance at a rate of 2.5 cm a 1 to balance trench retreat. This case would correspond to the Cascades (see A on Figure 4a). In the second case, the upper plate and the trench are fixed, and the neutral regime is reached when the subducting plate moved trenchward at a rate of 5 cm a 1. This case would correspond to the eastern Aleutians (see B on Figure 4a). The last case is a subducting plate moving trenchward at a rate of 10 cm a 1 and an upper plate retreating at 2.5 cm a 1. This combination implies that the trench advances at the same rate as the upper plate retreat, i.e., 2.5 cm a 1. This last case is similar to the northern Kurils (see C on Figure 4a). We thus see that the Figure 2. Correlation between the normal components of trench and subducting plate velocities in the HS3 reference frame for the 166 selected transects (see selection criteria in the main text). Fast subducting plates promote trench advance, whereas slow ones favor trench retreat. Regression line equation is V t = 0.62 V sub 3.7 in cm a 1 (quality factor R 2 = 0.37). 6of15

7 V up /V sub as well as the quality of the regression on neutral transects, though less than for HS3 (V up /V sub = 0.46, and R 2 = 0.37), is better for NNR (V up /V sub = 0.32, and R 2 = 0.23) than for SB04 (V up /V sub = 0.24, and R 2 = 0.11). We still observe that compression is observed in the upper plate only for combinations of plate velocities below the neutral line (V up < 0.3 V sub 1.9 for NNR and V up <0.2V sub 1.4 for SB04). The New Hebrides are still located in the compressional domain for NNR. Some other extensional transects, like Ryukyu and Tonga, overlap with the neutral domain. Figure 3. Overriding plate absolute motion V up versus subducting plate absolute motion V sub for the 166 selected transects. White dots are transects where slab age at trench is younger than 70 Ma, whereas black dots are for those older than 70 Ma. The diagram and the sketch inside the diagram are adapted from work by Heuret [2005]. neutral regime can be obtained with various combinations of plate and trench motions. [16] In Figure 6 with a few modern examples, we illustrate the evolution of upper plate deformation (v d )by varying the velocity of one of the converging plates while keeping the other one constant, i.e., an upper plate retreating at V up =1cma 1 (Figures 4a and 6a) and a subducting plate moving at a rate V sub = 2 cm a 1 (Figures 4a and 6b). Large variations of V sub (from 11 to 2 cm a 1 for V up =1cma 1 ) produce shortening (e.g., NE Japan) or spreading (e.g., Sandwich) in the upper plate, and small variations of V up (from 4 to1cma 1 for V sub =2cma 1 ) produce the same effect, i.e., shortening (e.g., Columbia) or spreading (e.g., Sandwich). [17] Returning to the dependency of the results on the reference frame, we observe in Figures 4c (NNR) and 4d (SB04) that both V up and V sub show lower variations in HS3 as a result of little or no net rotation. Strangely, the slope 3. Discussion [18] On the basis of this study, we observe that strain in the upper plate is predominantly controlled by its absolute motion [Heuret and Lallemand, 2005; Heuret et al., 2007]. Indeed, the slope V up /V sub of the neutral line is 0.5, meaning that V up is twice as influential on the tectonic regime of the upper plate as is V sub. V up thus dominates the control of back-arc stress but is not enough, as illustrated by NE Japan, which shows (in HS3) that a retreating upper plate can be counterbalanced by a fast subducting plate to produce back arc shortening. In other words, the kinematic conditions required to balance the arcward and oceanward forces applied along the interface between the plates with respect to the upper plate are satisfied when V up =0.5V sub 2.3, whenever the main upper plate (or the trench, because V t = V up when v d = 0) retreats or advances. Stress transmission and upper plate deformation occur when velocities deviate from this equilibrium. [19] The New Hebrides transects do not satisfy the general rule in any of the chosen reference frames, in the sense that the combination of velocities V up /V sub predict a compressional regime and not an extensional one. This is undoubtedly due to the unique plate configuration in this complex area [see Pelletier et al., 1998]; we have chosen the Pacific plate as the main upper plate, but the system is 3-D with several active spreading centers sandwiched between the Pacific plate to the north and the Australian plate to the west and southeast. This back-arc area is known for its multiridge activity caused by active upper mantle convection [Lagabrielle et al., 1997]. We thus consider that mantle flow may disturb plate interactions. [20] The Tonga subduction zone also warrants some discussion. The transects across this subduction zone fall into the extensional domain when using HS3 (Figure 4b) because the overriding Australian plate is retreating enough Figure 4. Same diagrams as in Figure 3: V up versus V sub, but divided into three groups depending on the tectonic regime within the upper plate (see Table 1). (a) Velocities in HS3 reference frame. The regression line V up =0.5V sub 2.3 in cm a 1 (or V sub 2 V up = 4.6) is valid only for the neutral subduction zone transects (white dots). Quality factor R 2 = Along the neutral line, the trench velocity V t equals the upper plate velocity V up because v d = 0. All transects characterized by active compression are located below this line, and all transects characterized by active extension are located above the line. (b) Luz, Luzon; Col, Columbia; C-Am, Central America; Phil, Philippines; Kur, Kurils; Kam, Kamtchatka; Ker, Kermadec; Izu, Izu-Bonin; Ryu, Ryukyu; Mar, Mariana; Ant, Antilles; Sand, Sandwich; Sum, Sumatra; Nan, Nankai; Ale, Aleutians; Casc, Cascades; Pat, Patagonia; Ala, Alaska; Anda, Andaman; Mak, Makran; N-Heb, New Hebrides; N-Brit, New Britain. (c) The same diagram in NNR reference frame. The regression line equation is V up = 0.32V sub 1.9. Quality factor R 2 = (d) The same diagram in SB04 reference frame. The regression line equation is V up = 0.24V sub 1.4. Quality factor R 2 = of15

8 Figure 4 8of15

9 Figure 4. to compensate the fast-moving Pacific plate. This is not true in the NNR or SB04 reference frames (see Figures 4c and 4d). As discussed by Heuret and Lallemand [2005], we consider this region (that includes the New Hebrides and Tonga arcs) to be subject to strong mantle flows, probably rising from active mantle convection beneath the North Fiji Basin [Lagabrielle et al., 1997]. It is therefore not surprising that the transects here often deviate from general subduction rules Spontaneous Trench Motion [21] The most informative result of this paper concerns the neutral domain where trench motion V t equals upper plate motion V up, which is approximately equivalent either to the absence of upper plate motion or to a trench motion that is not affected by the upper plate, at least in terms of stress (continued) transmission. In the case of the neutral regime (v d = 0), trench motion depends only on the balance of forces between the subducting plate and the surrounding mantle. In Figure 5, we illustrate that spontaneous trench motion may vary with rollback, fixity, and advance by changing V sub. Spontaneous trench motion is thus not only restricted to trench rollback, as suggested by Chase [1978] or Hamilton [2003]. This result is still valid in the NNR and SB04 reference frames, for which the Sunda trench (from Sumatra to Java) is advancing. Funiciello et al. [2004], Bellahsen et al. [2005], and Di Giuseppe et al. [2008] reached similar conclusions experimentally Interplate Coupling and Stress Transmission [22] Another result of this study involves the source of interplate coupling and stress transmission from one plate to Figure 5. Three specific examples taken along the neutral line where the deformation rate v d =0, i.e., trench velocity V t equals upper plate velocity V up. See location of transects on the neutral line in Figure 4a. (left) Cascades with a fixed subducting plate producing trench rollback at a rate of 2.5 cm a 1. (middle) Eastern Aleutians, upper plate (and thus trench) are fixed for a subducting plate moving at 5 cm a 1. (right) Northern Kurils with a fast subducting plate (10 cm a 1 ) inducing trench advance at a rate of 2.5 cm a 1. 9of15

10 Figure 6. (a) Three specific examples for a constant upper plate retreat of 1 cm a 1. Horizontal arrows indicate trench and slab motion. See Figure 1 for the definition of velocities and location of transects in Figure 4a. A dynamic equilibrium (neutral regime) is reached in Java. From the equation of kinematic equilibrium, we see that compression occurs when V sub increases (NE Japan), and extension occurs when V sub decreases (Sandwich). In the case of NE Japan, Pacific plate motion reaches 11 cm a 1. Back arc shortening occurs at a rate of about 1 cm a 1, so that the trench advances at 2 cm a 1, and the subduction rate is =11cma 1. (b) Three specific examples for a constant subducting plate trenchward advance of 2 cm a 1 (see location of transects in Figure 4a). In this case, dynamic equilibrium is reached when both plates advance trenchward at a rate of 2 cm a 1 (Cascades), compression occurs when V up increases (trenchward motion, Colombia), and extension occurs when V up decreases (arcward motion, Sandwich). the other. We show here that compressional stress does not simply increase with increasing convergence velocity v c but is also favored for moderate v c when V sub <0.5V up 2.3. This is even more true for extensional stress, which can even be reached when v c is large (e.g., v c =8cma 1 in Tonga), since V sub >0.5V up 2.3 remains true Forces Balance [23] A long list of authors so far addressed the dynamical aspects of subduction/convection system from kinematic data [e.g., Forsyth and Uyeda, 1975; McKenzie, 1977; Davies, 1980; Carlson, 1995; Conrad and Hager, 1999; Conrad and Lithgow-Bertelloni, 2002; Becker et al., 1999; Buffett and Rowley, 2006; Faccenna et al., 2007; Di Giuseppe et al., 2008], but the quantification of the different contributions in the subduction system remains elusive and dependent upon assumptions and somehow arbitrary scaling parameters. [24] The analysis presented here, although not pretending to give exact quantification of the forces at work, first uses the mode of trench migration as a proxy for the competition between the different contributions. The slab/mantle system is here considered to result from the competition between arcward and oceanward forces acting on a slab, or more specifically, the folding and unfolding torques acting at the pivot made by the slab hinge. To better understand the role of the main forces, we will assume a constant radius of curvature R c, slab length L, and dip a. We neglect the interaction of the slab with both the 660 km discontinuity and the lower mantle, which are extremely difficult to model because slab imagery at this depth is often unclear and difficult to interpret. We consider the change in slab dip to be negligible, so that trench motion equals horizontal slab migration. [25] The correlation between the age of the subducting plate A and its absolute motion V sub indicates that slab pull is the main force driving the motion of the subducting plate. Resisting forces are partitioned between mantle and lithosphere [Conrad and Hager, 1999]. We neglect friction and suction at the top of the slab since they are at least 1 order of magnitude smaller [Turcotte and Schubert, 1982; Conrad and Hager, 1999]. We assume that the upper mantle behaves passively as it is excited by slab subduction and migration. The asthenospheric mantle viscously resists to facewise translation of the slab, i.e., form drag. Finally, we distinguish three forces and related torques (see Figure 7). 10 of 15

11 Figure 7. Sketch showing the velocities, forces, torques, and other parameters used in the notations. [26] 1. The slab pull force F sp represents the main driving force in the system [e.g., Forsyth and Uyeda, 1975; McKenzie, 1977; Davies, 1980; Carlson, 1995]. It scales with plate thermal thickness T th, excess mass of the slab with respect to the mantle Dr, gravity acceleration g, and slab s length L, F sp ¼ 0:25DrgT th L ð1þ p T th ¼ 2:32 ffiffiffiffiffiffi ka ; ð2þ [27] [Turcotte and Schubert, 1982], k being the thermal diffusivity, and A being the slab s age in seconds. The coefficient 0.25 accounts for the decreasing of the excess mass of the slab with depth [Carlson et al., 1983]. F sp exerts a bending torque M sp at the pivot that contributes to the downward folding of the slab at the hinge. Except for buoyant subducting plates, this torque transmits an oceanward pressure along the interface between the plates. M sp is the product of F sp with the lever arm between the hinge and the gravity center of the slab (Figure 7), L M sp ¼ F sp cos a ð3þ 2 Keeping L and a constant, the bending torque M sp is proportional to the age of the subducting lithosphere at trench A since Dr is a direct function of T th. [28] 2. Within the lithosphere, the main resisting force is related to the bending of the lithosphere at trenches. We use an elastic rheology for the slab as a proxy because of the lack of constraints to adjust an efficient bending viscosity for a viscous slab at trench. For a semifinite elastic slab, the bending moment M b is the flexural rigidity of the plate D divided by its mean radius of curvature R b [Turcotte and Schubert, 1982], M b ¼ D R b ð4þ D ¼ ET ð eþ 3 12ð1 n 2 Þ ; ð5þ where E and n are the mean Young modulus and Poisson ratio, respectively, and pffiffiffi T e ¼ 4: A is the elastic thickness in meters with A (expressed in Ma) [see McNutt, 1984]. The torque M b resists the bending of the lithosphere and thus exerts an arcward pressure along the ð6þ 11 of 15

12 Figure 8. Sketch showing the balance of torques for the three examples illustrated in Figure 5. The slab pull torque M sp increases with the age of the subducting plate A, as does V sub if we consider that it is the main driving force for the subducting plate (see explanations in section 3.3). The mantle reaction to slab migration generates an anchoring torque M a which tends to unbend the plate when the trench rolls back and to bend it when the trench is advancing. interface between the plates. Assuming a constant R b and a constant a, the moment M b then increases with A 3/2. [29] 3. There is an additional force F a that accounts for the mantle reaction to slab facewise translation, including toroidal and poloidal flows. Some equations have been proposed by Dvorkin et al. [1993] or Scholz and Campos [1995] on the basis of the translation of an ellipsoid through a viscous fluid [Lamb, 1993]. In this study, we propose a simpler analytical solution adapted from work by Panton [1996] who considers the motion of an oblate spheroid (disk) into an infinite viscous fluid. The anchoring force F a is thus estimated as 6p 3 þ 2e F a ¼ ð Þ ah 5 m U N ; and the viscous shear force F sh is estimated as F sh ¼ 6pð4 þ eþ ah 5 m U T ; where e < 1 is the ratio between the disk radius a and its thickness taken as 1/4 if we consider a circular slab with L = 800 km and T th = 100 km. Here h m is the average viscosity of the displaced mantle, and U is the translation rate with normal and tangential components U N and U T with respect to the slab surface (see Figure 7), U N ¼ V t sin a U T ¼ n s þ V t cos a: ð7þ ð8þ ð9þ ð10þ [30] The viscous drag acting on both sides of the descending slab does not produce any torque at the slab hinge. We will thus only consider the anchoring force in this balance. To be compared with the first two forces, F a is normalized to the slab s width taken as equal to L, since we consider a circular slab in this equation. Here a should thus be replaced by a/l which is equal to 1/2. The simplified equation of the anchoring force thus becomes F a ¼ 6:6h m V t sin a; and the anchoring torque becomes ð11þ M a ¼ F a L 2 : ð12þ [31] For a given mantle viscosity h m, dip angle a, and slab length L, this torque scales with the facewise translation rate U N = V t sina. The force is directed either arcward or oceanward depending on the sense of slab motion, since it resists slab forward or backward motion. [32] Assuming no stress transmission through the plates interface (no significant deformation of the supposedly weak arc), we balance the oceanward/folding and arcward/unfolding torque components of these three forces (Figure 8), taking into account that in steady state, the sum of the moments should be zero. We write that M sp ¼ M b þ M a ; ð13þ M a being positive in the case of slab rollback, null for a fixed trench, and negative in the case of an advancing slab. These situations correspond to low, moderate, and high V sub, respectively. The three examples in Figure 8 correspond to those illustrated in Figure 5. [33] Let us consider three subduction systems (Figure 8) with the same length L = 700 km, dip a =50, and radius of curvature of subducting slabs R b = 350 km and values for 12 of 15

13 Table 3. Parameters Used in the Calculations and Notations a Quantity Symbol Value (Range) Thermal diffusivity k 10 6 m 2 s 1 Gravity acceleration g 9.81 m s 2 Young modulus E 155 GPa Poisson ratio n 0.28 Asthenosphere viscosity h m Pa s Crustal density r c 2900 kg m 3 Lithospheric mantle density r lm 3300 kg m 3 Asthenosphere density r a 3230 kg m 3 a Young modulus and Poisson ratio are given for a range of subducting lithospheres aged 10 to 140 Ma [Meissner, 1986]. the other parameters as listed in Table 3. We compare in Table 4 the behavior of five cases, including the three ones mentioned in Figure 8, with various ages of subducting lithospheres, 20, 60, 80, 100 and 140 Ma and respective trench absolute velocities V t, 2.5, 0, +2.5 and +5 cm a 1. According to Cloos [1993], we can estimate the bulk density of an oceanic plate from those of a 7 km thick oceanic crust and a lithospheric mantle whose thickness is deduced from the total thermal thickness T th (see equation (2)). This allows us to calculate the excess mass Dr for each slab, taking mean densities for the crust, lithospheric mantle, and asthenosphere as given in Table 3. [34] We observe in Table 4 that M b is lower than M sp for young slabs, except the very young ones for which M sp can become negative, and exceeds M sp for old ones. Using our values (see Table 3), the change happens for slabs aged about 80 Ma for which M b ffi M sp. Slab pull forces F sp vary from 2.2 to N per unit length of trench, leading to a slab pull moment M sp from 0.5 to N m in the range of slabs aged between 20 and 140 Ma at trench, whereas the bending moment M b varies from 0.3 to N m. The anchoring force F a both depends on slab facewise translation velocity V t sina and mantle viscosity h m. Table 4 is thus only indicative of tendencies but should be adapted depending on some key parameters like h m, which is a matter of debate between low values, to Pa s, and higher ones, to Pa s [Winder and Peacock, 2001; Cadek and Fleitout, 2003; Mitrovica and Forte, 2004; Enns et al., 2005]. If we use instead of Pa s, then our anchoring moments will increase by a factor of 10. All other parameters may also vary, like L, a, R b, Dr, or the slab width. Ultimately, we have chosen to consider a passive mantle that opposes to slab facewise migration, but we have evidence of mantle flow in various regions, especially around slab lateral edges, as in Tonga. In this case, we must count with an additional force exerted by the mantle onto the slab. [35] From the balance of forces and earlier observations, we can say that arcward trench motion (advancing) is marked by high resistance of the plate to bending at the trench together with a quickly subducting plate, and that the opposite occurs for trench rollback. This explains the negative correlation between the age of the lithosphere at trench A and V t [Heuret and Lallemand, 2005], i.e., trench rollback for young subducting plates and vice versa together with the positive correlation between A and V sub (Figure 3). Such observations highlight the role of the bending resistance that scales with the cube of the square root of A, whereas the slab pull scales with A only. It is therefore dominant for old (and fast) subducting plates [Conrad and Hager, 1999; Becker et al., 1999; Faccenna et al., 2007]. Di Giuseppe et al. [2008] also pointed out the competition between slab stiffness and slab pull. They concluded that advancing-style subduction is promoted by thick plate, a large viscosity ratio, or a small density contrast between plate and mantle. On the basis of 2-D numerical models with a viscosity structure constrained by laboratory experiments for the deformation of olivine, Billen and Hirth [2007] also concluded that the slab stiffness has a stronger affect on dynamics than small differences in slab density due to plate age. [36] In our model, we have pointed out how the lithosphere and the mantle together control trench migration. We consider that the classification of transects into three groups (extensional, neutral, and compressional) as a function of V sub and V up is another positive argument in favor of the HS3 reference frame. [37] One important point is that back arc rifting or spreading is not necessarily related to trench rollback, even in the SB04 reference frame (see Izu-Bonin, Kermadec, or Andaman). These behaviors are the result of a specific combination of V sub and V up that favors either extension or compression. 4. Conclusions [38] By selecting modern subduction zones for which the overriding plates do not deform significantly, we have explored the kinematics of trenches with respect to both Table 4. Estimates of Forces and Related Torques for Various Slab Ages and Trench Migration Rates a A (Ma) Te (km) T th (km) Dr (kg m 3 ) D (10 22 Nm) v t (cm a 1 ) F sp (10 12 N) F a (10 12 N) M sp (10 18 Nm) M a (10 18 Nm) M b (10 18 Nm) Trench rollback Fixed trench Advancing trench Advancing trench Advancing trench a See section 2 for the definition of parameters and Table 3 for the values used in the calculations. We have used h m =10 20 Pa s. Analogies can be found with modern subduction zones such as Cascades (A = 10 Ma, rollback, V t = 2.5 cm a 1, a =45 ), Luzon (A = 20 Ma, rollback, V t = 8 cma 1, a =70 ), eastern Aleutian (A = 60 Ma, advance, V t = 0.5 cm a 1, a =60 ), Java (A = 80 Ma, advance, V t = 1.5 cm a 1, a =70 ), Kamtchatka (A = 100 Ma, advance, V t =2cma 1, a =60 ), northern Kuril (A = 110 Ma, advance, V t =2cma 1, a =50 ), and Izu-Bonin (A = 140 Ma, advance, V t =5cma 1, a =70 ). 13 of 15

14 the behaviors of the subducting plate (rheology and kinematics) and the reaction of the mantle. We have shown that neutral subduction zones satisfy a combination of velocities: V sub 2V up = 4.6 cm a 1 in the HS3 reference frame. For higher V sub, trenches advance faster than the upper plates retreat, and compression is favored, whereas for lower V sub, trenches retreat faster than the upper plates advance, favoring extension. In the neutral domain, in which V t = V up,we have shown that spontaneous trench motion can occur either oceanward (rollback) for slow (and young) subducting plates or arcward (advance) for fast (and old) subducting plates. We implicitly treat the paradox of trench rollback that generally occurs with young slabs but not with old ones. We propose that this behavior is mainly caused by the resistance of the subducting plate to bending, as its stiffness (as well as its velocity) increases with age faster than the slab pull for plates older than 80 Ma. [39] Acknowledgments. This research was supported by the CNRS- INSU DyETI program Dynamics of Subduction and as part of the Eurohorcs/ESF-European Young Investigators Awards Scheme, by funds from the National Research Council of Italy and other national funding agencies participating in the Third Memorandum of Understanding, as well as from the EC Sixth Framework Programme. We thank all our colleagues who participated in this program for the numerous discussions and debates and especially Neil Ribe and Jean Chéry. We also thank Richard Carlson and Wouter Schellart for their improvement of an earlier version of this paper, Thorsten Nagel and an anonymous reviewer for their constructive review, and Anne Delplanque for her precious help in line drawings. References Becker, T. W. (2006), On the effect of temperature and strain-rate dependent viscosity on global mantle flow, net rotation, and plate-driving forces, Geophys. J. Int., 167, , doi: /j x x. 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